Electrons do not "know" their exact radius where they will emit a photon. The behavior of electrons and the emission of photons are described by quantum mechanics, which is a mathematical framework that provides probabilities and statistical descriptions for the behavior of particles at the microscopic level.
In quantum mechanics, electrons are described by wave functions, which are mathematical functions that contain information about the probability distribution of finding the electron in different locations. The square of the wave function, known as the probability density, gives the probability of finding the electron at a particular location.
When an electron transitions from a higher energy state to a lower energy state within an atom or a material, it can emit a photon. The probability of this emission occurring at a particular location is related to the shape of the wave function and the energy difference between the initial and final states. The wave function of the electron provides information about the probability distribution of finding the electron at different distances from the nucleus, but it does not determine an exact radius for photon emission.
The emission of a photon by an electron occurs through quantum processes involving interactions with the electromagnetic field. The exact position and timing of photon emission are inherently uncertain due to the probabilistic nature of quantum mechanics. The emission of a photon can be described in terms of probabilities and statistical distributions, rather than deterministic rules.
In summary, the emission of photons by electrons is governed by the probabilistic nature of quantum mechanics, and the exact radius of emission cannot be determined with certainty. The behavior of electrons and photons is described by mathematical models that provide probabilities and statistical distributions for their interactions.